Transcript lec06_heapsort

Heapsort
• O(n lg n) worst case
• Another design paradigm
– Use of a data structure (heap) to manage information during
execution of algorithm
• Comparision-based Sorting Algorithm
Analysis of Algorithms
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Heap Data Structure
Analysis of Algorithms
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Heap Property
Analysis of Algorithms
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A Heap Example
Analysis of Algorithms
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Heap Data Structure
Analysis of Algorithms
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Heap Operations
Analysis of Algorithms
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Heap Operations
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Maintaining Heap
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Runtime Analysis of HEAPIFY
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Formal Runtime Analysis of HEAPIFY
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Formal Runtime Analysis of HEAPIFY
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Building Heap
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Proof of Lemma
Analysis of Algorithms
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Building Heap
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Build Heap - Example
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Build Heap - Example
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BuildHeap – Tighter Running Time Analysis
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BuildHeap – Tighter Running Time Analysis
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BuildHeap – Tighter Running Time Analysis
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BuildHeap – Tighter Running Time Analysis
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HEAPSORT
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HEAPSORT
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Heapsort - Example
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Heapsort - Example
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Heapsort - Example
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Heapsort – Running Time Analysis
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How fast can we sort?
• The comparison sorting algorithms such as insertion sort,
mergesort, quicksort and heapsort only use comparisons to
determine the relative order of elements.
• The best worst-case running time that we’ve seen for
comparison sorting is O(n lg n) .
Is O(n lg n) the best we can do?
• Decision trees can help us answer this question.
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Decision-tree example
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Decision-tree example
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Decision-tree model
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Lower bound for decision-tree sorting
Corollary. Heapsort and merge sort are asymptotically
optimal comparison sorting algorithms.
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